Persistent membranous cross correlations due to the multiplicity of gates in ion channels

J Comput Neurosci (2011) 31:713–724DOI 10.1007/s10827-011-0337-9

Persistent membranous cross correlationsdue to the multiplicity of gates in ion channels

Marifi Güler

Received: 27 November 2010 / Revised: 18 April 2011 / Accepted: 25 April 2011 / Published online: 17 May 2011© Springer Science+Business Media, LLC 2011

Abstract Ion channels in excitable cells revealspontaneous intermittent opening and closing. As themembrane area reduces, this stochasticity enablesspontaneous firing and elevates the cell’s ability tofire at weaker stimuli. A multiple number of gates areaccommodated in each individual ion channel. Here weinvestigate the possible impact of that gate multiplicityon the cell’s function specifically when the membranearea is of limited size. It is shown that a non-triviallypersistent correlation then takes place between thetransmembrane voltage fluctuations (also between thefluctuations in the gating variables) and the componentof open channel fluctuations attributed to the abovegate multiplicity. This cross correlation persistency isfound to be playing a major augmentative role in theelevation of the cell’s excitability and spontaneousfiring; without the persistency, the cell would be muchless excitable. The cross correlation persistency is alsofound to enhance spike coherence. The stochasticHodgkin–Huxley equations, put forward by Fox andLu, are addressed in the context of their recognizedfailure to produce accurate enough statistics of spikegeneration. Our results indicate that the major sourceof that inaccuracy is the incapability of the stochastic

Action Editor: Carson C. Chow

M. Güler (B)Department of Computer Engineering,Eastern Mediterranean University,Famagusta, via Mersin-10, Turkeye-mail: [email protected]

Hodgkin–Huxley description to reflect the above crosscorrelation persistency.

Keywords Ion channel · Channel gate ·Channel noise · Excitability · Small size membrane ·Stochastic neuronal dynamics

1 Introduction

Excitability of cells is a fundamental physiological pro-cess wherein the conductance of potassium and sodiumis facilitated by voltage-gated ion channels (Hille 2001).Cellular activity is under the influence of the noise ofexternal and internal types—for an overview see e.g.Faisal et al. (2008). Internal noise, contrary to theexternal type of noise that arises from synaptic trans-mission and network effects, is specific to neuronsand generates stochastic behavior on the level of neu-ronal dynamics. The major source of internal noise isdue to the existence of a finite number of voltage-gated ion channels in a patch of neuronal membrane.These channels have one open state and one or moreclosed states. The number of open channels fluctuatesin a seemingly random manner (Sakmann and Neher1995) implying a fluctuation in the conductivity of themembrane, which in turn, implies a fluctuation in thetransmembrane voltage. When the membrane area isvery large, i.e. when the number of ion channels is big,the voltage dynamics is described by the celebratedHodgkin and Huxley (1952) equations. For smallermembrane patches, however, the effects of the conduc-tance fluctuations on the voltage activity of the cell arepotentially profound and not negligible. Experimentalinvestigations suggest that spike patterns are directly

714 J Comput Neurosci (2011) 31:713–724

affected by the stochasticity of single channel open-ings (Sigworth 1980; Lynch and Barry 1989; Johanssonand Arhem 1994), and that ion channel noise can causespontaneous firing (Koch 1999; White et al. 1998).Patch-clamp experiments in vitro have demonstratedthat channel noise in the soma and in the dendritesproduces voltage fluctuations that are large enough toaffect timing, initiation, and, propagation of action po-tentials (Diba et al. 2004; Jacobson et al. 2005; Dorvaland White 2005; Kole et al. 2006). The phenomenoncalled stochastic resonance has been observed to occurin a system of voltage-dependent ion channels formedby the peptide alamethicin (Bezrukov and Vodyanoy1995).

Numerical simulations of channel dynamics andtheoretical investigations have shown internal noisefrom ion channels to cause spontaneous activity (inthe form of repetitive spiking or bursting) in otherwisequiet membrane patches (DeFelice and Isaac 1992;Strassberg and DeFelice 1993; Fox and Lu 1994; Chowand White 1996; Rowat and Elson 2004; Güler 2007,2008); furthermore, have revealed the occurrence ofstochastic resonance and the coherence of the gener-ated spike trains (Jung and Shuai 2001; Schmid et al.2001; Özer 2006). Even in the presence of largenumbers of ion channels, channel fluctuations canbecome critical near the action potential threshold(Schneidman et al. 1998; Rubinstein 1995); the timingprecision of an action potential is determined by thesmall number of ion channels that are open at theaction potential threshold. It has also been shown thatspike propagation in axons is affected from ion channelnoise (Faisal and Laughlin 2007; Ochab-Marcinek et al.2009).

Our recent theoretical work (Güler 2007) indicatesthat not only the fluctuations in the number of opengates, but also so-called the renormalization effects,induced by the presence of a multiple number of gatesin individual ion channels, might well be playing asignificant role in the cell’s activity when the membranearea is of limited size. The main aim of the presentpaper was to determine if the gate multiplicity in ionchannels truly induces any such peculiar effects, and,if so, to comprehend the underlying mechanism fromthe first principles. We essentially find that the gatemultiplicity gives rise to unprecedented persistent crosscorrelations in the small size membranes, and that thispersistency plays a key role for the pronounced eleva-tion of the cell’s excitability. The main findings of thepaper are characterized both by theoretical argumentsand numerical simulations of channel dynamics.

2 The membrane dynamics

The transmembrane voltage V evolves in accordancewith

CdVdt

= −gKψK (V − EK) − gNaψNa (V − ENa)

−gL (V − EL) + I (1)

where the dynamic variable ψK is the proportion ofopen potassium channels to the total number of potas-sium channels in the membrane; similarly, ψNa is theproportion of open sodium channels. The values ofthe constant membrane parameters used in Eq. (1)are given in Table 1. In the Hodgkin–Huxley (HH)equations, the channel variables ψK and ψNa were ap-proximated to their deterministic values ψK = n4 andψNa = m3h; since there exist four n-gates in a potas-sium channel, together with three m-gates and one h-gate in a sodium channel. All the gates in a channelmust be open for the channel to open. Here, n, m andh are the gating variables. Let NK and NNa denotethe total numbers of potassium and sodium channels,respectively. Then 4NKn, 3NNam, and NNah give thenumbers of open n-gates, open m-gates, and open h-gates, correspondingly.

The following Markov process applies for the dy-namics of gates. If an n-gate is closed at time t, thenthe probability that it remains closed at time t + �t isexp(−αn�t), and if it is open at time t, then the proba-bility that it remains open at time t + �t is exp(−βn�t).The parameters αn and βn are voltage-dependent open-ing and closing rates of n-gates. An analogous proce-dure applies to the m-gates and the h-gates. The ratefunctions adopted read as

αn = (0.1 − 0.01V)/(exp(1 − 0.1V) − 1) (2a)

βn = 0.125 exp(−V/80) (2b)

Table 1 Constants of the membrane

C Membrane capacitance 1 μF/cm2

gK Maximal potassium conductance 36 mS/cm2

EK Potassium reversal potential −12 mVgNa Maximal sodium conductance 120 mS/cm2

ENa Sodium reversal potential 115 mVgL Leakage conductance 0.3 mS/cm2

EL Leakage reversal potential 10.6 mVDensity of potassium channels 18 chns/μm2

Density of sodium channels 60 chns/μm2

J Comput Neurosci (2011) 31:713–724 715

αm = (2.5 − 0.1V)/(exp(2.5 − 0.1V) − 1) (2c)

βm = 4 exp(−V/18) (2d)

αh = 0.07 exp(−V/20) (2e)

βh = 1/(exp(3 − 0.1V) + 1). (2f)

3 The non-trivial cross correlation persistency (NCCP)

3.1 NCCP attributed to the potassium channels

Since there is more than one n-gate in a potassiumchannel, knowing the proportion of open gates, n, doesnot suffice to specify ψK uniquely. For example, a toymembrane comprised of just two potassium channels(eight gates) may be in a state, at time t2, that oneof the channels has four open gates and the other hastwo; while in a state, at time t1, that each channel hasthree open gates. Then, even though the membranehas the same number of open gates at both times,one channel is open at t2 while no channel is openat t1—see Fig. 1. Let us coin the term gate-to-channeluncertainty to describe this lack of knowledge that takesplace in ψK even if n is known; and the term gate noiseto denote the random fluctuations in n. A differentterminology was used in our earlier works. We changedit here in respect to some colleagues who have found

t1

t2

Fig. 1 Two possible conformational states of a toy membrane,at two different times t1 and t2. The membrane is comprised ofjust two potassium channels (eight n-gates). Filled black dots andsmall circles represent open and closed gates, respectively. Thebigger circles represent channels. Despite the numbers of opengates at t1 and at t2 being the same (six), one channel (shadowed)is open at t2 while no channel is open at t1—the gate-to-channeluncertainty

the original terminology not expressly clear enough.The gate-to-channel uncertainty reveals itself as dy-namic random fluctuations in the construct ψK − [ψK].The construct singles out the channel fluctuations thatarise from the gate-to-channel uncertainty. It wouldvanish, irrespective of the gate noise, if the gate-to-channel uncertainty were not present. Here [ψK] standsfor the configuration average of the proportion ofopen potassium channels; computed over the possibleconfigurations of the membrane having 4NKn open n-gates. It is not difficult to evaluate that

[ψK]= (4NKn−3)(4NKn−2)(4NKn−1)n(4NK−3)(4NK−2)(4NK−1)

, if NKn≥1

(3)

and [ψK] = 0, otherwise. Hence the construct ψK −[ψK] measures how much the number of open channelsdiffer from the configuration average at any given in-stant. Unless the membrane is extremely small, it holdsthat [ψK] ≈ n4. In the limit of infinite membrane size,the fluctuations in the construct vanish and, therefore,for very large membranes we can use the HH valueψK = [ψK] = n4 at all times. If each channel had asingle gate the construct would be irrelevant for anymembrane size, as we would identically have ψK =[ψK] = n then.

Let us define order parameters �VK and �n

K as thefollowing cross correlations:

�VK =

⟨(ψK − [ψK])V

⟩−

⟨ψK − [ψK]

⟩⟨V

⟩⟨[ψK]

⟩(ENa − EK)

(4a)

�nK =

⟨(ψK − [ψK])n

⟩−

⟨ψK − [ψK]

⟩⟨n⟩

⟨[ψK]

⟩⟨n⟩ (4b)

where the expectation values 〈· · · 〉 are the ensemble av-erages over the conformational states of the membrane.Every conformational state in the ensemble evolves intime, independently of the others, via the Markovianevolution of the constituting gate states and Eq. (1);this way the ensemble at time t + �t is decided fromthe ensemble at time t. The terms in the denominatorswere included simply for the convenience of scalingand dimensionality. �V

K is a measure of the correlationbetween the fluctuations of the construct ψK − [ψK]and the fluctuations of the voltage V. �n

K is similar, butemploys the fluctuations of n rather than of V. Sincethe positivity or negativity of the construct is totally


random and cannot be influenced by the fluctuationsof V or of n, one may take for granted that the orderparameters decay to zero after an initial transient pe-riod. We however show, both by numerical simulationof channels here and by theoretical arguments in thenext section, that this intuition fails. For the numericalevaluation, we employ the method known as the simplestochastic method—see e.g. Zeng and Jung (2004)—in which all the gates are simulated individually usingthe Markov scheme in Section 2. Once the numbersof open channels are known, the voltage is integratedto the next time of gate update using Eq. (1). Thecomputation of the order parameters, for an exemplarmembrane patch, reads as in Fig. 2 when the inputcurrent is I = −4 μA/cm2. The reason for using such asmall external input current stems from the fact that weare interested in the properties of the order parameterswithin the phase of subthreshold activity. We observefrom the figure that both �V

K and �nK acquire non-

zero values and remain always negative. That is, for thenear-equilibrium dynamics, a non-transient correlationtakes place between the fluctuations of the constructψK − [ψK] and the fluctuations of V, and also of n.Note that �V

K and �nK oscillate in Fig. 2, rather than

staying constant in time. This is because of the use ofan inevitably finite ensemble in the simulation. Theamplitudes of oscillations decrease steadily with theincrease in the ensemble size.

3.2 NCCP attributed to the sodium channels

The construct that reveals the gate-to-channel uncer-tainty associated with the sodium channels is ψNa −

-0.003

-0.002

-0.001

150 300 450

Time (ms)

0

ΩKV

ΩKn

Fig. 2 The order parameters �VK and �n

K . A membrane patch of1,800 potassium channels and 6,000 sodium channels was usedin the computation. The input current is I = −4 μA/cm2. Anensemble of 8,000 evolving conformational membrane states wasused in the simulation

[ψNa]. Here the configuration average of the proportionof open sodium channels, [ψNa], reads as

[ψNa] = (3NNam − 2)(3NNam − 1)m(3NNa − 2)(3NNa − 1)

h, if NNam ≥ 1

(5)

and [ψNa] = 0, otherwise. In the limit of infinitemembrane size, [ψNa] yields its corresponding HHvalue m3h. Then the order parameters of concern aregiven by

�VNa =

⟨(ψNa − [ψNa])V

⟩−

⟨ψNa − [ψNa]

⟩⟨V

⟩⟨[ψNa]

⟩(ENa − EK)

(6a)

�mNa =

⟨(ψNa − [ψNa])m

⟩−

⟨ψNa − [ψNa]

⟩⟨m

⟩⟨[ψNa]

⟩⟨m

⟩ (6b)

�hNa =

⟨(ψNa − [ψNa])h

⟩−

⟨ψNa − [ψNa]

⟩⟨h⟩

⟨[ψNa]

⟩⟨h⟩ . (6c)

The computation of the order parameters �VNa and

�mNa, for an exemplar membrane patch, reads as in

Figs. 3 and 4, respectively, when the input currentis I = −4 μA/cm2. The order parameter �h

Na is notplotted here as it was found to attain a value very closeto zero. Undermining the oscillations arising from thefiniteness of the used ensemble, it is seen that both�V

Na and �mNa attain non-zero values and remain always

positive. The algebraic sign of the construct ψNa −[ψNa] is totally random and cannot be influenced by the

0.001

0.002

150 300 450

ΩN

aV

Time (ms)

0

Fig. 3 The order parameter �VNa. A membrane patch of 1,800

potassium channels and 6,000 sodium channels was used in thecomputation. The input current is I = −4 μA/cm2. An ensembleof 8,000 evolving conformational membrane states was used inthe simulation


-0.01

0.01

0.02

150 300 450

ΩN

am

Time (ms)

0

Fig. 4 The order parameter �mNa. A membrane patch of 1,800

potassium channels and 6,000 sodium channels was used in thecomputation. The input current is I = −4 μA/cm2. An ensembleof 8,000 evolving conformational membrane states was used inthe simulation

fluctuations of V, or of m, or of h; but somehow, a non-trivially non-transient correlation takes place betweenthe fluctuations of the construct and the fluctuations ofV, and also of m.

4 Analysis of NCCP

Next we attempt to explicate how the order parameterscan remain non-zero.

4.1 The persistence of �VK and �n

K

The state of a gate at time t + �t is dependent onthe state at time t; although the degree of depen-dence decreases with the increase in the value of �t,as specified by the Markovian transition probabilities.This implies that the construct ψK − [ψK] has a non-vanishing autocorrelation function: the autocorrelationtime is finite, but not zero. Making use of the positivityof V − EK, it can be deduced from Eq. (1) that ifψK − [ψK] > 0 throughout some period of time then anegative variation in dV/dt occurs along that period.Here the variation is relative to the case of having ψK −[ψK] = 0 over the same period. This, in turn, entailsthat a negative variation in V also materializes in thatperiod. To depict that property we write

ψK − [ψK] > 0 ⇒ δ(dV

dt

)< 0 ⇒ δV < 0. (7)

Similarly, the variation in the case of negative ψK −[ψK] is depicted by

ψK − [ψK] < 0 ⇒ δ(dV

dt

)> 0 ⇒ δV > 0. (8)

It follows from Eqs. (7) and (8) that, irrespective ofthe sign of ψK − [ψK], the product (ψK − [ψK])δV isnegative at every moment of a time period through-out which ψK − [ψK] does not switch sign. A pictorialexplanation is provided in Fig. 5 when the residencetime of ψK − [ψK] in the same algebraic sign is notshorter than the duration of a typical fluctuation inV. On the other hand, if ψK − [ψK] switches sign atsome point within the period then the product willnot be negative at every moment; for a brief intervalright after the sign switch it will be positive. If it isassumed that the above residence time is notably longerthan the relaxation time of the product to return tonegative again, the probability of finding the productto be negative will be higher than the probability offinding it positive. Therefore, the fluctuations of V willbecome negatively correlated with the fluctuations ofψK − [ψK]. Consequently, the order parameter �V

K willattain a negative value. The occurrence of a variationδV in V, with the deviations from ψK − [ψK] = 0, is alsothe reason for the order parameter �n

K taking a non-zero value. The rates αn and βn are voltage-dependentfunctions; the former increases with the voltage, andthe latter decreases—see Fig. 6. Since an increase inαn reduces the probability of a closed n-gate remainingclosed, and, a decrease in βn raises the probability ofan open n-gate remaining open, a positive δV causesa positive variation in the gating variable n. This way,like �V

K, �nK also attains a negative value. Note that

if ψK − [ψK] had a vanishing autocorrelation function,contrary to the actuality, �V

K and �nK would then van-

ish. If the autocorrelation time were zero, ψK − [ψK]

Time

<V>

V ψK - [ψK]

+0-

δV > 0

δV < 0

Fig. 5 An illustration of the variation in the voltage V, in re-sponse to deviations of the construct ψK − [ψK] from zero. Thevariation is denoted by δV. The sign of ψK − [ψK] was assumedto remain the same throughout the duration of a typical voltagefluctuation


0.05

0.09

0.13

-10 -5 0 5 10 15

Voltage (mV)

αn(V)

βn(V)

Fig. 6 The rate functions αn and βn plotted against the voltage,using Eqs. (2a) and (2b)

would be able to switch its sign instantaneously inan infinitesimally small time interval, and, successivesign switches will crop up within a duration not anylonger than the relaxation time. Then ψK − [ψK] andδV will not be correlated; and rapid sign switches ofthe construct will suppress the growth of |δV|. In otherwords, the property of the autocorrelation time beinggreater than zero is vital to the revelation of NCCP.

4.2 The persistence of �VNa and �m

Na

How �VNa and �m

Na can attain non-zero values can be ex-plained along the same lines as for the order parameters�V

K and �nK. Making use of the negativity of V − ENa,

it can be deduced from Eq. (1) that if ψNa − [ψNa] > 0over some period of time then a positive variation indV/dt arises in comparison with the case where ψNa −[ψNa] = 0. This, in turn, entails that a positive variationin V also materializes in that period. That is

ψNa − [ψNa] > 0 ⇒ δ(dV

dt

)> 0 ⇒ δV > 0. (9)

Similarly, the variation in the case of negative ψNa −[ψNa] is depicted by

ψNa − [ψNa] < 0 ⇒ δ(dV

dt

)< 0 ⇒ δV < 0. (10)

Assuming that the residence time of ψNa − [ψNa] in thesame algebraic sign is not shorter than the duration ofa typical fluctuation in V, it then follows from Eqs. (9)and (10) that the product (ψNa − [ψNa])δV is positive,irrespective of the sign of ψNa − [ψNa]; see Fig. 7 for apictorial depiction. If it is assumed that the residencetime of ψNa − [ψNa] in the same algebraic sign is long

Time

<V>

V ψNa - [ψNa]

-0+

δV > 0

δV < 0

Fig. 7 An illustration of the variation in the voltage V, in re-sponse to deviations of the construct ψNa − [ψNa] from zero. Thevariation is denoted by δV. The sign of ψNa − [ψNa] was assumedto remain the same throughout the duration of a typical voltagefluctuation

enough, the probability of finding the product to bepositive will be higher than the probability of finding itnegative. Therefore the fluctuations of V will becomepositively correlated with the fluctuations of ψNa −[ψNa]. Consequently, the order parameter �V

Na willattain a positive value. The occurrence of a variationδV in V, with the deviations from ψNa − [ψNa] = 0, isalso the reason for the order parameter �m

Na taking anon-zero value. The rate αm increases with the voltage,and βm decreases—see Fig. 8. Since an increase in αm

reduces the probability of a closed m-gate remainingclosed, and, a decrease in βm raises the probability ofan open m-gate remaining open, a positive δV causesa positive variation in the gating variable m. This way,like �V

Na, also �mNa attains a positive value. The order

parameter �hNa can attain a value only very near to

1

4

7

-10 -5 0 5 10 15

Voltage (mV)

αm(V)

βm(V)

Fig. 8 The rate functions αm and βm plotted against the voltage,using Eqs. (2c) and (2d)


zero since αh and βh, compared to αm and βm, changevery slowly with V. If ψNa − [ψNa] had a vanishingautocorrelation function, contrary to the actuality, �V

Naand �m

Na would then vanish.

5 NCCP elevates the cell’s excitability and the spikecoherence

The analysis conducted above indicates that NCCPmodulates voltage fluctuations. Consider the sketchgiven in Fig. 5. When ψK − [ψK] > 0, smaller volt-age values are favored by the fluctuations, in com-parison with the voltage values in the case of ψK −[ψK] = 0. When ψK − [ψK] < 0, larger voltage valuesare favored. The voltage fluctuations are also modu-lated in the same manner by the persistency inducedby the multiplicity of m-gates in sodium channels—see Fig. 7. Consequently, we anticipate NCCP to in-crease diversity in the amplitudes of subthreshold volt-age fluctuations by increasing the probability of largerfluctuation amplitudes. In order to be able to assessthis effect by simulations, we fabricate an operation;call it as gate shuf f ling. The operation alters the actualconformation of the membrane by shuffling the loca-tions of the gates randomly, following the update ofthe gate states, at each time step. The shuffling appliesto both the n-gates and the m-gates, but to each kindseparately. Although the shuffling applied at a giventime step does not change the number of open gatesat this particular step, it can easily vary the number ofopen channels due to the gate-to-channel uncertainty,and therefore can affect the future dynamics. For exam-ple, a toy membrane comprised of just two potassiumchannels may be in a state, at some given time, wherethere are three open gates in each channel; six in total.After the gate shuffling, the total number of open gateswill be still six, but the conformation now may be thatone of the channels has four open gates and the otherhas two. That is, the gate shuffling has transformedthe membrane from having no open channel to havingone channel open. The gate shuffling causes the auto-correlation time of the construct ψK − [ψK], and alsoof the construct ψNa − [ψNa], to be zero. Then, withreference to our earlier argument, the order parametersmust vanish when the gate shuffling is applied; wehave found by simulations that it is indeed the case.Hence, applying the gate shuffling to the gate dynamicsremoves NCCP, as it was intended to do. Therefore,for the inspection of the role of NCCP on the voltagefluctuations, it just suffices to compare the two voltagetime series corresponding to: (a) the actual dynamics,(b) the dynamics subjected to the gate shuffling. Such

-6

-4

-2

200 500 800

Volt

age

(mV

)

Time (ms)

(A)

200 500 800

-6

-4

-2

Time (ms)

(B)

Fig. 9 Subthreshold voltage fluctuations. A membrane patch of1,800 potassium channels and 6,000 sodium channels was used inthe computation. The input current is I = −4 μA/cm2. Result (a)is from the dynamics that uses open gates at their actual locations.Result (b) is from the dynamics that uses the locations of the opengates as subjected to the gate shuffling

a comparison has been carried out and is presented inFig. 9. It is seen that NCCP can drastically increase theamplitudes of the subthreshold voltage fluctuations tovalues much larger than those caused by the gate noisealone. Using the same membrane patch and the sameinput current value as in Fig. 9, we have measured thestandard deviation of the voltage to be about 0.7 mV forthe actual dynamics but about 0.17 mV for the dynam-ics with the gate shuffling. One might mistakenly tendto think at this point that the primary reason for observ-ing a reduction in the amplitudes of voltage fluctuationsin Fig. 9(b) may not be the removal of NCCP, but ratherthat the gate shuffling somehow causes a reduction inthe amplitudes of fluctuations in the constructs ψK −[ψK] and ψNa − [ψNa]. But, as demonstrated in Fig. 10,the gate shuffling operation just eliminates the auto-correlation in the fluctuations of ψK − [ψK], withoutaffecting the amplitudes of fluctuations in the construct;therefore, the operation only removes NCCP with noother direct effects. Moreover, we will show below thatthe spike statistics are not sensitive to the amplitudes ofconstruct fluctuations when the autocorrelation in thefluctuations is destroyed.

-0.003

0

0.004

205 227 250

ψK

- [

ψK

]

Time (ms)

(A)

205 227 250

-0.003

0

0.004

Time (ms)

(B)

Fig. 10 Fluctuations of the construct ψK − [ψK] at subthreshold.A membrane patch of 1,800 potassium channels and 6,000 sodiumchannels was used in the computation. The input current is I =−4 μA/cm2. Result (a) is from the dynamics that uses open gatesat their actual locations. Result (b) is from the dynamics that usesthe locations of the open gates as subjected to the gate shuffling


Since NCCP causes an increase in the amplitudesof subthreshold voltage fluctuations, it should facilitatethe cell’s spiking by making the transition from thesubthreshold phase to the firing phase easier. To showthe significance of this facilitation, we provide somespike statistics in Figs. 11, 12, 13 for various membranesizes. The statistics contain the mean spiking rates,in a comparative manner, corresponding to: (a) theactual dynamics, (b) the dynamics subjected to the gateshuffling, (c) the dynamics where it was set to ψK =n4 and ψNa = m3h. In the dynamics (c), the channelvariables are set to their deterministic values, but thethe gating variables are decided in the same way as in(a) and (b). Since the channel variables are uniquelydetermined by the the gating variables in the dynamics(c), the gate-to-channel uncertainty does not hold forthis dynamics and the noise in it is just the gate noise.Firstly, we observe that the gate shuffled dynamics canfire—with an increasing probability as the membranesize is reduced—even in some range of input currentvalues in which the deterministic HH dynamics fails tofire; the HH dynamics needs I � 6.3 μA/cm2 to suc-ceed firing. Second, the dynamics (b) and (c) give verysimilar statistics; that is the gate noise plays no effective

0

10

20

30

40

50

60 70

-2 0 2 4 6 8

Mea

n s

pik

ing

rat

e (H

z)

Input current (μA/cm2)

A) Actual

B) Shuffled

C) ΨK=n4,ΨNa = m3h

Fig. 11 Mean spiking rates against the input current, displayedby a membrane patch comprised of 360 potassium channelsand 1,200 sodium channels. The three plots shown correspondto: (a) the actual dynamics, (b) the dynamics subjected to thegate shuffling, (c) the dynamics where it was set to ψK = n4

and ψNa = m3h. The averages were computed over a 30 s timewindow

0

10

20

30

40

50

60

0 2 4 6 8

Mea

n s

pik

ing

rat

e (H

z)


A) Actual

B) Shuffled


Fig. 12 Mean spiking rates against the input current, displayedby a membrane patch comprised of 1,800 potassium channelsand 6,000 sodium channels. The three plots shown correspondto: (a) the actual dynamics, (b) the dynamics subjected to thegate shuffling, (c) the dynamics where it was set to ψK = n4


role on the spiking activity when NCCP is eliminated.But the striking observation is that the actual dynam-ics, compared to the gate shuffled dynamics and thedynamics (c), can fire at much smaller input currentvalues together with significantly higher spiking ratesin the current range I < 6.3 μA/cm2. That is to say,NCCP elevates the excitability. Figure 13 demonstratesthat NCCP continues to facilitate spontaneous firingeven in a range of larges membrane sizes where thegate noise alone falls short of activating the cell. Thusfinite size membranes owe their elevated excitabilitynot only to the gate noise but, to a greater extent, also toNCCP.

NCCP may also have an effect on the coherence ofspike trains. A measure of the spike coherence is thecoefficient of variation given by

√⟨T2

⟩ − ⟨T

⟩2⟨T

⟩

where⟨T

⟩and

⟨T2

⟩are the mean and mean-squared

interspike intervals, respectively. The coefficient of var-iation computations, corresponding to the actual dyna-mics and the dynamics subjected to the gate shuffling,


0

10

20

30

40

50

60

1 2 3 4 5 6 7 8

Mea

n s

pik

ing

rat

e (H

z)


A) Actual

B) Shuffled


Fig. 13 Mean spiking rates against the input current, displayedby a membrane patch comprised of 7,200 potassium channelsand 24,000 sodium channels. The three plots shown correspondto: (a) the actual dynamics, (b) the dynamics subjected to thegate shuffling, (c) the dynamics where it was set to ψK = n4


are presented in Fig. 14 for an exemplar membranepatch. It is seen that the actual dynamics exhibitssmaller values of the coefficient of variation than thegate shuffled dynamics in the range of input currentswhere both dynamics are capable of firing—see Fig. 12for the firing range. Thus NCCP enhances the coher-ence of spiking.

6 A discussion on the implications for the stochasticHodgkin–Huxley equations

HH equations have been extended by Fox and Lu(1994) into stochastic differential equations to includethe effects of ion channel noise in neuronal dynamics.In this widely used diffusion approximation to the dis-crete gate dynamics, the equations of neuronal activitydiffer from HH equations only by the presence of an ad-ditional voltage-dependent mean zero Gaussian whitenoise term in the Langevin equation for each gatingvariable. It reads therein that

n = αn(1 − n) − βnn + ηn(t) (11a)

m = αm(1 − m) − βmm + ηm(t) (11b)

h = αh(1 − h) − βhh + ηh(t) (11c)

where the noise terms have the variances⟨ηn(t)ηn

(t′) ⟩

= 1

NK[αn(1 − n) + βnn] δ

(t − t′

)(12a)

⟨ηm(t)ηm

(t′) ⟩

= 1

NNa[αm(1−m)+βmm] δ

(t−t′

)(12b)

⟨ηh(t)ηh

(t′) ⟩

= 1

NNa

[αh(1−h)+βhh

]δ(t−t′

). (12c)

Simulations however revealed that the stochasticHH description fails to produce accurate enough sta-tistics of spike generation in general (Mino et al. 2002;Zeng and Jung 2004; Bruce 2009; Sengupta et al. 2010).These studies suggest that the Langevin model of Foxand Lu may not be suitable for accurately simulatingchannel noise, even in simulations with large numbersof ion channels. Our present work provides a tangibleexplanation for this failure. Although the gate noise—or the gate shuffled dynamics—can be modeled simplyby introducing some appropriate white noise terms ofvanishing means into the deterministic HH equationsas in Eq. (11), NCCP cannot be done in the same way.Since, in the stochastic HH description, the channelvariables ψK and ψNa are uniquely decided by the

0

0.2

0.4

0.6

0.8

1

1.2

1.4

5 6 7 8 9 10 11


A) Actual B) Shuffled

Co

eff,

of

vari

atio

n

Fig. 14 The coefficient of variation against the input current,displayed by a membrane patch comprised of 1,800 potassiumchannels and 6,000 sodium channels. (a) the actual dynamics, (b)the dynamics subjected to the gate shuffling. The averages werecomputed over a 30 s time window


gating variables, through ψK = n4 and ψNa = m3h—the same as in the HH equations—the description lacksthe capability of reflecting NCCP; no matter whetherthe noise terms in Eq. (11) are white or colored,Gaussian or non-Gaussian, of mean zero or not. Onemay argue that the effect of NCCP making the cellmore excitable can be in essence obtained by havinga noise variance in Eq. (11) larger than that neededto model the gate noise; this is what the Fox and Luformulation to some extent does. Fox and Lu employsa stochastic automaton model of the gates to derivemaster equations for potassium and sodium channelsusing four n-gates in a potassium channel, and, threem-gates and one h-gate in a sodium channel. For thederivation of the Langevin approximation from themaster equations, they, however, considered the potas-sium channels to be made up of a single element oftype n and the sodium channels to be made up of alsoa single element, but this time with two types, the mtype and the h type. Following the derivation of theLangevin noise variances in Eq. (12), they raised thechannel variables to the appropriate powers, ψK = n4

and ψNa = m3h, for inclusion into the conductances.Thus the noise variance given by Eq. (12c) correspondsto the variance needed to model the gate noise of h-gates; but, the noise variance given by Eq. (12a) is fourtimes of the variance needed to model the gate noiseof n-gates, and, the noise variance given by Eq. (12b)is three times of the variance needed to model the gatenoise of m-gates. Having, in the Langevin Eqs. (11a)and (11b), noise terms with variances larger than thatneeded to model the gate noise, results in the gatingvariables being too noisy and can therefore only be asomewhat rough substitute for the function of NCCP.For relatively larger membrane sizes, on the otherhand, the noise terms might lose their role while NCCPis still intact, because the weakening in the effect ofNCCP with increasing membrane size is at a slowerrate than the weakening in the effect of gate noise.Stochastic HH equations, therefore, must incorporatesome new voltage-dependent functional forms (or somecolored noise terms) into the conductances, in additionto the white noise terms accommodated in the Langevinequations to model the gate noise. Then we articulatethe following renormalization of the channel variables

ψK ≈ n4 + φK(V − Veq(I), 1/NK, σ 2

K

)(13a)

ψNa ≈ m3h + h φNa(V − Veq(I), 1/NNa, σ

2Na

)(13b)

where φK and φNa are some functions such that φK → 0in the limit of NK → ∞, and, φNa → 0 in the limitof NNa → ∞. The input current dependent function

Veq(I) corresponds to the voltage value at the exactequilibrium obtained from the steady state solution ofHH equations. At V = Veq(I), both φK and φNa vanish.σ 2

K is the configuration variance of the open potassiumchannels, computed over the possible configurations ofthe membrane having 4NKn open n-gates. Similarly,σ 2

Na is the configuration variance of the open sodiumchannels, computed over the possible configurationsof the membrane having 3NNam open m-gates. Thescheme (13) apparently makes the order parametersyield non-zero values, and, therefore, integrates NCCPinto the formulation. The question of what functionalforms reflect NCCP most faithfully, remains open.Some preliminary findings that we have obtained usingan adopted first order renormalization scheme indicatethat the renormalized HH equations give a lot moreaccurate results than the Fox and Lu formulation. Letus report the details in a subsequent article as work onthe issue is still in progress.

7 Concluding remarks

In this paper, we showed that the presence of a multiplenumber of gates in individual ion channels plays aremarkable role in the cell’s function when the mem-brane area is of limited size; the multiplicity inducesnon-trivially persistent membranous cross correlations,which turns out to be the major cause of the upriseof excitability in small size neuronal membranes. Wefound that NCCP continues to facilitate spontaneousfiring even in a range of relatively large membrane sizeswhere the gate noise alone falls short of activating thecell. We also found that NCCP enhances the coherencein the spike trains. We argued that the dynamics ofsize limited cells cannot be modelled, beyond a lim-ited accuracy, purely by introducing some white noiseterms of vanishing means into the deterministic HHequations; in addition, it is necessary to renormalizethe channel variables in HH equations into some newvoltage-dependent functional forms.

The necessity of renormalization corrections inthe equations of activity was argued in our earlierwork (Güler 2007) in the context of stochastic renor-malization, using a tailored membrane obeying theRose–Hindmarsh dynamics in the limit of infinite mem-brane size. The present work shows that such a renor-malization is genuine, and has NCCP at its root. Therenormalization corrections were found to cause be-havioral transition from quiescence to spiking in somerange of input currents (Güler 2008); essentially thesame effect as NCCP is seen to be exerting. This isan indication that the incidence of the renormalization


correction terms is a universal phenomenon that is tooccur irrespective of the underlying membrane model.It was found that the presence of renormalization cor-rections can lead to faster temporal synchronizationof the respective discharges of electrically coupled twoRose–Hindmarsh type neuronal units (Jibril and Güler2009). Therefore, NCCP should be expected to be play-ing a facilitative role in temporal synchronization ofsynaptically coupled neurons, and worth being investi-gated. Moreover, the role of NCCP in the case of time-varying and noisy input currents per se deserves to beinvestigated, with reference to stochastic resonance.

Added note While our manuscript was under review,an article (Linaro et al. 2011), that proposes an alterna-tive to the Fox and Lu’s Langevin equations, was pub-lished. The formulation therein uses the appropriatepowers of the deterministic gating variables, rather thanthe stochastic gating variables, in deciding the propor-tions of open channels. But some Ornstein–Uhlenbeckprocesses, with the diffusions obtained from the covari-ances of n4 and m3h, accompany the conductances. Theformulation appears to yield better spiking statisticsthan the Fox and Lu equations, but some discrepanciesfrom the actual spiking statistics still occur.

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Persistent membranous cross correlations due to the multiplicity of gates in ion channels

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